A class of maximal operators with rough kernel on product spaces (Q5954278)

Consider the class of singular integral operators NEWLINE\[NEWLINET_K f(x,y)= \int^\infty_0 \int^\infty_0 \iint_{S^{n-1}\times S^{m-1}} K(r\xi, s\eta) f(x- r\xi, y-s\eta) r^{n-1} s^{n-1} d\xi d\eta dr ds,NEWLINE\]NEWLINE where \(K\) is a kernel of the form \(K(r\xi,s\eta)= r^{-n} s^{-m} \sum_j a_j(r, s)\Omega_j(\xi, \eta)\) with \(\Omega_j\in L^q(S^{n-1} \times S^{m-1})\), \(1< q\leq\infty\), \(\Omega_j(t\xi, s\eta)= \Omega_j(\xi,\eta)\) for any \(t,s> 0\), \(\int_{S^{n-1}} \Omega_j(\xi,\eta) d\xi= 0\) for any \(\eta\in S^{n-1}\), and \(\int_{S^{m-1}} \Omega_j(\xi,\eta) d\xi= 0\) for any \(\xi\in S^{m-1}\).NEWLINENEWLINENEWLINELet \(M\) and \(q'\) denote by the class of all kernels \(K\) and the conjugate index of \(q\), respectively. The authors proved the theorem that if \(q\) and \(p\) satisfy one of the following conditions: NEWLINE\[NEWLINE1< q< 2\quad\text{and}\quad \max\{2nq'/(2n+nq'- 2), 2mq'/(2m+ mq'- 2)\}< p< 2q'/(q'- 2),\tag{a}NEWLINE\]NEWLINE NEWLINE\[NEWLINE2\leq q\leq \max\{2(n- 1)/(n- 2), 2(m-1)/(m-2)\}\quad\text{and}\tag{b}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\max\{2nq'/(2n+ nq'- 2), 2mq'/(2m+ mq'- 2)\}< p<\infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINEq> \max\{2(n-1)/(n-2), 2(m-1)/(m- 2)\}\tag{c}NEWLINE\]NEWLINE and \(1< p<\infty\), then the maximal operator \(\sup_{K\in M}|T_k f|\) is bounded on \(L^p(\mathbb{R}^{n-1}\times \mathbb{R}^{m-1})\). This result is an extension of \textit{L.-K. Chen} and \textit{X. Wang} [J. Math. Anal. Appl. 164, No. 1, 1-8 (1992; Zbl 0764.42011)] to product spaces.